Generation of out-of-plane polarized spin current by spin swapping

The generation of spin currents and their application to the manipulation of magnetic states is fundamental to spintronics. Of particular interest are chiral antiferromagnets that exhibit properties typical of ferromagnetic materials even though they have negligible magnetization. Here, we report the generation of a robust spin current with both in-plane and out-of-plane spin polarization in epitaxial thin films of the chiral antiferromagnet Mn3Sn in proximity to permalloy thin layers. By employing temperature-dependent spin-torque ferromagnetic resonance, we find that the chiral antiferromagnetic structure of Mn3Sn is responsible for an in-plane polarized spin current that is generated from the interior of the Mn3Sn layer and whose temperature dependence follows that of this layer’s antiferromagnetic order. On the other hand, the out-of-plane polarized spin current is unrelated to the chiral antiferromagnetic structure and is instead the result of scattering from the Mn3Sn/permalloy interface. We substantiate the later conclusion by performing studies with several other non-magnetic metals all of which are found to exhibit out-of-plane polarized spin currents arising from the spin swapping effect.


I. Structural analysis of Mn3Sn/Py, Cu/Py, Ru/Py, Re/Py and Pt/Py films
To analyze the crystal symmetry of Mn3Sn, X-ray diffraction experiments were carried out by using a Gallium-Jet X-ray source operated at 70 keV and 100 W emitting Ga-K radiation (=1.3414 Å) monochromatized and focused onto the sample by a Montel optics providing a highly collimated beam of 100 µm in height and 2 mm in width. For data collection a six-circle x-ray diffractometer operated in the z-axis mode 1 was used, where the incident beam was kept constant at an angle of µ=2 to enhance the scattered intensity of the 12 nm thick Mn3Sn film by simultaneously penetrating the 5 nm Py and 3 nm TaN layers. Integrated reflection intensities were measured by rotating the sample around the surface normal (thetascan) while the position of the 2-dimensional (2D) pixel detector was kept fixed at an in-plane () and out-of-plane () angle given for each reflection (HKIL) 1 .
The data set for Mn3Sn film consists of about 22 reflections which reduce to 12 symmetry independent reflections after averaging over symmetry equivalent reflections based on the 6mm symmetry of the (0001) sample surface. Subsequently, the observed structure factor magnitudes, |F(HKL)obs|, were derived from the integrated intensities by multiplying with instrumental factors (Lorentz, polarization-and effective area) [2][3] . In the first step of the structure analysis, the bulk structure model was considered to fit the calculated |F(HKL)calc| to the observed ones |F(HKL)obs|. The structure model is shown in Fig. 1(a) in projection along the [0001] axis (main text). Bulk Mn3Sn is reported to crystallize in the inversion symmetric space group (SGR) P63/mmc (PDF 01-073-2857) 4 , in which Sn and Mn atoms occupy Wyckoff sites 2d ( 1 /3, 2 /3, ¾) and 6h (x, 2x, 1 /4), respectively. Owing to the high symmetry of the structure, the only free parameter is the x-position of the Mn-atoms and the atomic displacement parameters (ADP) of the Sn-and Mn atoms 5 . All observed reflections are in agreement with the general condition for their appearance [(H H 2H ̅ L): L=2n, with n integer].
In the first step of the quantitative analysis the calculated |F(HKL)| were fitted to the experimental ones based on SGR P63/mmc using the least-squares refinement based on nonweighted refinement which minimizes the non-weighted residuum ( u ) given by: , where the summation runs over all reflections (HKL). For the Mn3Sn film, we derive u =0.190 (see Table 1). In the next step, the Fourier-Transform (FT) of the structure factors was calculated to derive the three-dimensional charge density [(x,y,z)] within the unit cell, which is given by: (x, y, z) = ∑ ±| ( ) | cos 2 ( x + y + z). Since the structure models are inversion-symmetric the experimental structure factors as coefficients of the Fourier series are real numbers. Thus, only the signs in front of the structure factor magnitudes [F(HKL)= ± |F(HKL)|] are needed for the calculation. They are calculated based on the structure model. As long as its gross features are correct (as in the present case) the FT is able to provide a direct view of the structure and helps to identify unknown details. In Fig. S1(a) the charge density of the 12 nm film is shown in projection along [0001]. In comparison with the bulk structure model in Fig. 1(a) (main text), it is directly evident that the intense maxima are linked to the Sn and Mn-atoms. The distinct difference between the samples is that [ Fig. S1(a)] there is another maximum at the origin of the unit cell (x,y)=(0,0) (see arrow).
The structure refinement involving a fractional occupation of the site labelled by the Wyckoff notation (2b) in SGR P63/mmc dramatically improved the fit as shown by the contour plot in Fig. S1(b). We emphasize that there is another faint charge density at ( 1 /2, 1 /2, z), which however does not lead to a significant improvement in the fit quality. and Al2O3 substrates ( Fig. S3(a,b)), respectively. The surface roughness is found to be less than in Fig. S4(a) for Mn3Sn/Py. Further cross-sectional transmission microscopy reveals that Mn3Sn is well-ordered and confirms the thicknesses of all the layers ( Fig. S4(b)).

II. Magnetic properties of Mn3Sn and exchange bias in Mn3Sn/Py thin films
Magnetization (M) as a function of in-plane magnetic field ( ∥ ) has been measured on a single layer of 12 nm Mn3Sn (0001) film to probe the small in-plane moment which lies in the kagome plane. From the M vs. ∥ data, a small in-plane magnetization (~20 emu/cm 3 ) at 300 K is observed when the diamagnetic contribution is subtracted from the substrate ( Fig.   S5(a)). The small magnetization is consistent with the previous report on the thin film 6 . Further, magnetization decreases with an increase in temperature which indicates the Néel temperature beyond 400 K (Inset of Fig. S5(a)). A detailed exchange bias (EB) measurement has been performed on the Mn3Sn/Py thin films to confirm the antiferromagnetic structure of Mn3Sn.
The Mn3Sn/Py film was cooled from 400 K to the measurement temperature in the presence of a negative in-plane magnetic field ( ∥ = −10 kOe) and magnetic hysteresis was recorded at the respective temperatures. A large shift of the magnetic hysteresis loop along the positive magnetic field axis has been observed (inset of Fig. S5(b)) at 5 K. This shift decreases with an increase in the temperature and a symmetric hysteresis (i.e., EB vanishes) has been observed at 280 K which is the blocking temperature (TB) for the Mn3Sn (12 nm)/Py (5 nm) film ( Fig.   S5(b)). The blocking temperature decreases systematically with a decrease in Mn3Sn layer thickness (Fig. S5(c)). Note that since the TB is less than 300 K for all the bilayers, EB does not affect the SOT measurements at 300 K.

III. Longitudinal resistivity and anomalous Hall effect of Mn3Sn thin film
Figure S6(a) shows zero-field longitudinal resistivity ( ) as a function of temperature over the temperature range 5 K to 400 K for the 12 nm Mn3Sn film. It exhibits a metallic behaviour with a residual resistivity ratio (RRR) ~ 1.6 which manifests a high-quality thin film. Note that, Ru resistivity is subtracted to calculate the for Mn3Sn film using a parallel resistor model. We have also investigated Hall resistivity ( ) as a function of outof-plane magnetic field (Hz // (0001)) at room temperature for the same film ( Fig. S6(b)). As expected from symmetry when Hz // (0001) it does not show anomalous Hall effect 7 .  are shown in Fig. S7(a) for Mn3Sn (12 nm)/Py(5 nm) structure. mix are fitted using Eq.1.

IV. Effective magnetization and Gilbert damping constant in Mn3Sn/Py films
(Main text) and, extracted res and Δ . Frequency (f) as a function of res is fitted ( Fig.   S7(b)) using the Kittel formula (Eq. S1) to calculate the effective magnetization ( eff ) .
Similarly, Δ as a function of 'f' is fitted (Fig. S7(c)) using Eq. S2 to extract the Gilbert damping constant ( ). eff and are 774 emu/cm 3

V. Angular variation of and for different thicknesses of Mn3Sn layer
The S ( ) and A ( ) for the Mn3Sn(dAFM = 3-9 nm)/Py(5 nm) structures show similar angular dependence (Fig. S8(a-f)) like Mn3Sn(12 nm)/Py(5 nm) film ( Fig. 3(a-b) in main text). These results show that z,AD , z,FL do not depend on dAFM. Also, the addition of spin-pumping contribution, which shows sin( ) angular dependence 8 , to y,AD does not reproduce S ( ) . This rules out the spin-pumping contribution for all the films. The asymmetric angular variation of A further confirms the presence of z . The spin-torque efficiency 9-10 due to y (y) and z (z) as a function of dAFM are shown in fig. S8 (g,h). Both y and z show a weak dAFM dependence.

VI. Theory and spin Hall conductivity tensor for Mn3Sn
In linear response theory, the spin current with spin polarization along induced by the electric field is given as

VII. SOT after setting magnetic domain in the presence of an external magnetic field
To investigate how the large external magnetic field influences y and z , we have applied Hset =  7 T to set a specific spin texture of Mn3Sn before performing the ST-FMR experiments ( Fig. 1e upper panel, main text) and then measured the angular dependence of S and A of Mn3Sn(12 nm)/Py(5 nm) at 300 K. The angular dependence remains unchanged for 0 device (Fig. S9(a-d)) compared to the pristine films ( Fig. 3(a-d),

VIII. Spin-orbit torques for different crystallographic directions of Mn3Sn
We have measured SOTs for  = 0°, 45° and 90°. Note that = 0° represents the device where RF is along the in-plane crystallographic direction [011 ̅ 0] of Mn3Sn. The torque due to y ( y,AD ′ ) is independent of the crystal orientation/in-plane device angle whereas torques due to z ( z,AD ′ and z,FL ′ ) show a small angular dependence (Fig. S10 (a-c)). for Cu(5 nm)/Py(5 nm) film. mix are fitted using Eq.1. and extracted res and Δ .
Frequency (f) as a function of res is fitted (Fig. S11(b)) using the Kittel formula (Eq. S1) to calculate the effective magnetization ( eff ). Δ as a function of 'f' is fitted (Fig. S11(c)) using Eq. S2 to extract the Gilbert damping constant ( ). eff and are 843 emu/cm 3 Fig. S12(a) for Ru(5 nm)/Py(5 nm) film. Using Eq.1, mix are fitted and, res and Δ are extracted as fit parameters. Frequency (f) as a function of res is fitted (Fig. S12(b)) using the Kittel formula (Eq. S1) to calculate the effective magnetization ( eff ). Δ as a function of 'f' is fitted (Fig. S12(c)) using Eq. S2 to extract the Gilbert damping constant ( ). eff and are 887 emu/cm 3 and 0.007 respectively. Here also, S ( ) and A ( ) show the presence of y,AD , y,FL , z,AD and z,FL (Fig. S12(d-e)) and angular variation shows a sign change as spin Hall angle of Ru is opposite to Cu or Pt. for Re(5 nm)/Py(5 nm) film. mix are fitted using Eq.1. and extracted res and Δ .
Frequency (f) as a function of res is fitted (Fig. S13 (b)) using the Kittel formula (Eq. S1) to calculate the effective magnetization ( eff ). Δ as a function of 'f' is fitted (Fig. S13 (c)) using Eq. S2 to extract the Gilbert damping factor ( ). eff and are 835 emu/cm 3

XII. Effect of Cu and Pt insertion in between Mn3Sn and Py
Here, we have investigated the SOT after inserting 2 nm Cu in between Mn3Sn and Py layer. This helps to break the interface between Mn3Sn and Py. The angular dependence of S and A for 0 device in Mn3Sn(12 nm)/Cu(2 nm)/Py(5 nm) shows similar behavior ( Fig.   S14(a-b)) like Mn3Sn(12 nm)/Py(5 nm) film ( Fig. 3(a-b), Main text) which shows a robust z after Cu insertion too. Further, we have measured the SOTs for 0 device after inserting 2 nm Pt in between Mn3Sn and Py layer which also shows the presence of y,AD , y,FL , z,AD and z,FL (Fig. S14(c-d)). The increase in the magnitude of S and A originates from the finite y,AD and y,FL of the Pt layer. This clearly shows that the interface near to Py is important for the generation of z . The magnitudes of ′ y,AD , ′ z,AD , ′ z,FL are 0.12 (0.28), 0.023 (0.031), 0.026 (0.04) for Cu (Pt) insertion, respectively.

XIII. Spin-orbit torque in poly-crystalline Mn3Sn films
We have grown polycrystalline 20 nm thick Mn3Sn on Al2O3 substrate without a Ru buffer layer. The anti-damping ( , ) torque due to y is not present and S () shows a very unusual angular variation (Fig. S15a). It is worth noting here that the epitaxial Mn3Sn (0002) film does not show an AHE whereas polycrystalline Mn3Sn exhibits a large AHE which makes the analysis more complicated. On the other hand, the angular dependence of A indicates the presence of a , ( ′ , = 0.028) due to z and Oersted field dominating field-like torque (Fig. S15b). We believe that there are two reasons which might affect the z . Firstly, from structural analysis, we find that the magnetic properties of Py films grown on polycrystalline Mn3Sn are different, e.g higher coercivity. Secondly, we find that the interface roughness increases significantly for growth on polycrystalline Mn3Sn thin films as compared to epitaxial

XIV. SOTs in Cu/Py and Pt/Py bilayers on Si/SiO2 and Al2O3
Angular dependence of S and A for Cu(5 nm)/Py(5 nm) and Pt(5 nm)/Py (5 nm) bilayers which are grown on Si(001)/SiO2(25 nm) substrate shows that z,AD and y,FL ( Fig.   S16(a,b,e,f) are very small compared to the same structures grown on Al2O3(0001) substrate ( Fig. S16(c,d,g,h). The better crystallinity of Cu/Py and Pt/Py on Al2O3 substrate which is evident from XRD (Fig. S3), is responsible for the large z,AD and z,FL . This clearly demonstrates that the spin-swapping mechanism dominates when the disorder is less.

XV. Comparison of SOTs in GSG and GS-type device
The shape and magnitude of S () and A () are strongly depends on the device geometry. Besides torques due to y , we observed a finite , and , torques in GSG device whereas only significant , dominated in GS devices 11

XVI. Spin-orbit torques in Cu/Fe and Cu/Ni bilayers
The theoretical 12 and experimental 13 developments of the orbital Hall effect (OHE) show only the existence of a y polarization which strongly depends on the magnetization of the ferromagnets. If OHE was the mechanism in our devices then z should strongly depend on the magnetization of the ferromagnet. Thus, we have made similar devices of Cu/Py where the Py layer was replaced by Fe or Ni. However, we see a very weak magnetization dependence of z (reduced by half from Cu/Fe to Cu/Ni) although we do find that y (reduced 4 times from Cu/Fe to Cu/Ni) strongly depends on the magnetization (see Fig. S18). This rules out the OHE as the origin of z and supports the spin-swapping origin of z . Note that in spin swapping mechanism, z weakly depends on the magnetization of the ferromagnets 14 .